0016–2663/04/3801–0067 c 2004 Plenum Publishing Corporation 67 Functional Analysis and Its Applications, Vol. 38, No. 1, pp. 67–68, 2004 Translated from Funktsional  nyi Analiz i Ego Prilozheniya, Vol. 38, No. 1, pp. 82–84, 2004 Original Russian Text Copyright c  by A. V. Kosyak Quasi-Invariant Measures and Irreducible Representations of the Inductive Limit of Special Linear Groups ∗ A. V. Kosyak Received December 18, 2002 Abstract. Unitary representations of the group G = SL 0 (2∞, R) = lim −→n SL(2n − 1, R) are con- structed. The construction uses G-quasi-invariant measures on some G-spaces that are subspaces of the space Mat(2∞, R) of two-way infinite real matrices. We give a criterion for the irreducibility of these representations. Key words: infinite-dimensional special linear group, irreducible unitary representation, quasi- invariant measure, Ismagilov’s conjecture. 1. Let X be a measurable space, α : G → Aut(X ) a measurable action of a group G on X , and µ a G-quasi-invariant measure on X . Then one can define a unitary representation π α,µ,X : G → U (L 2 (X, dµ)) of G by the formula (π α,µ,X t f )(x)=(dµ(α t −1 (x))/dµ(x)) 1/2 f (α t −1 (x)), f ∈ L 2 (X, dµ). Let α(G)= {α t ∈ Aut(X ) | t ∈ G}, and let α(G)  be the centralizer of the subgroup α(G) in Aut(X ), i.e., α(G)  = {g ∈ Aut(X ) |{g,α t } = gα t g −1 α −1 t = e for all t ∈ G}. Conjecture [1]. The representation π α,µ,X : G → U (L 2 (X, dµ)) is irreducible if and only if 1) µ g ⊥ µ for all g ∈ α(G)  \{e} ; 2) the measure µ is G-ergodic. This conjecture, which generalizes Ismagilov’s conjecture concerning the irreducibility of regular representations of infinite-dimensional groups (see [1] and references therein), was proved in [2] for the group G = B N 0 of finite upper triangular matrices of infinite order, some left coset spaces of the form X = G 0 \ B N , where G 0 is a subgroup of the group B N of upper triangular infinite-order matrices with ones on the main diagonal, and arbitrary Gaussian product measures on X . 2. Example. We claim that the conjecture holds for the group G = SL 0 (2∞, R), some G-spaces X that are subspaces of the space Mat(2∞, R) of real matrices with two-way infinite rows and columns, and arbitrary Gaussian product measures on X . Consider the space R 2∞ =(x n ) n∈Z of two-way infinite rows. We define Y m as the direct product of m copies ( m ∈ N) of R 2∞ . Let E kn ∈ Mat(2∞, R), k,n ∈ Z, be the matrix whose (k,n)th entry is equal to one and all other entries are zero. It is convenient to realize Y m as a subspace of Mat(2∞, R): Y m =  x ∈ Mat(2∞, R)    x = m  k=1  n∈Z x kn E kn  . The group G = SL 0 (2∞, R) = lim −→n SL(2n − 1, R) is the inductive limit of the groups G n = SL(2n − 1, R), n ∈ N, with respect to the embeddings G n  x → x + E −(n+1),−(n+1) + E n+1,n+1 ∈ G n+1 ,n ∈ N. Obviously, the formula R t (x)= xt −1 , t ∈ G, x ∈ Y m , specifies a well-defined right action of G = SL 0 (2∞, R) on Y m . We define a noncentered Gaussian product measure µ m (b,a) on Y m by the formula dµ m (b,a) (x)=  m k=1  n∈Z dµ kn (x kn ), where dµ kn (x kn )=(b kn /π) 1/2 exp(−b kn (x kn − a kn ) 2 ) dx kn , b =(b kn ), b kn > 0, and a =(a kn ), a kn ∈ R 1 ,1  k  m, n ∈ Z. Let T R,µ,m = π R,µ m ,Y m . If X = Y m , then the group α(G)  ⊂ Aut(X ) obviously contains the image of ∗ Supported in part by DFG grant 436 UKR 113/72.